Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{5k^2 + 85k + 350}{-9k^2 - 180k - 900}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {5(k^2 + 17k + 70)} {-9(k^2 + 20k + 100)} $ $ y = -\dfrac{5}{9} \cdot \dfrac{k^2 + 17k + 70}{k^2 + 20k + 100} $ Next factor the numerator and denominator. $ y = - \dfrac{5}{9} \cdot \dfrac{(k + 10)(k + 7)}{(k + 10)(k + 10)}$ Assuming $k \neq -10$ , we can cancel the $k + 10$ $ y = - \dfrac{5}{9} \cdot \dfrac{k + 7}{k + 10}$ Therefore: $ y = \dfrac{ -5(k + 7)}{ 9(k + 10)}$, $k \neq -10$